3.199 \(\int \frac{x}{3+6 x+2 x^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+3\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+3\right ) \]

[Out]

((1 - Sqrt[3])*Log[3 - Sqrt[3] + 2*x])/4 + ((1 + Sqrt[3])*Log[3 + Sqrt[3] + 2*x])/4

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Rubi [A]  time = 0.0155641, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {632, 31} \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+3\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(3 + 6*x + 2*x^2),x]

[Out]

((1 - Sqrt[3])*Log[3 - Sqrt[3] + 2*x])/4 + ((1 + Sqrt[3])*Log[3 + Sqrt[3] + 2*x])/4

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x}{3+6 x+2 x^2} \, dx &=\frac{1}{2} \left (1-\sqrt{3}\right ) \int \frac{1}{3-\sqrt{3}+2 x} \, dx+\frac{1}{2} \left (1+\sqrt{3}\right ) \int \frac{1}{3+\sqrt{3}+2 x} \, dx\\ &=\frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (3-\sqrt{3}+2 x\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (3+\sqrt{3}+2 x\right )\\ \end{align*}

Mathematica [A]  time = 0.02062, size = 44, normalized size = 0.9 \[ \frac{1}{4} \left (\left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+3\right )-\left (\sqrt{3}-1\right ) \log \left (-2 x+\sqrt{3}-3\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(3 + 6*x + 2*x^2),x]

[Out]

(-((-1 + Sqrt[3])*Log[-3 + Sqrt[3] - 2*x]) + (1 + Sqrt[3])*Log[3 + Sqrt[3] + 2*x])/4

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Maple [A]  time = 0.003, size = 31, normalized size = 0.6 \begin{align*}{\frac{\ln \left ( 2\,{x}^{2}+6\,x+3 \right ) }{4}}+{\frac{\sqrt{3}}{2}{\it Artanh} \left ({\frac{ \left ( 6+4\,x \right ) \sqrt{3}}{6}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(2*x^2+6*x+3),x)

[Out]

1/4*ln(2*x^2+6*x+3)+1/2*3^(1/2)*arctanh(1/6*(6+4*x)*3^(1/2))

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Maxima [A]  time = 1.40655, size = 55, normalized size = 1.12 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3} + 3}{2 \, x + \sqrt{3} + 3}\right ) + \frac{1}{4} \, \log \left (2 \, x^{2} + 6 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2*x^2+6*x+3),x, algorithm="maxima")

[Out]

-1/4*sqrt(3)*log((2*x - sqrt(3) + 3)/(2*x + sqrt(3) + 3)) + 1/4*log(2*x^2 + 6*x + 3)

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Fricas [A]  time = 1.6719, size = 136, normalized size = 2.78 \begin{align*} \frac{1}{4} \, \sqrt{3} \log \left (\frac{2 \, x^{2} + \sqrt{3}{\left (2 \, x + 3\right )} + 6 \, x + 6}{2 \, x^{2} + 6 \, x + 3}\right ) + \frac{1}{4} \, \log \left (2 \, x^{2} + 6 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2*x^2+6*x+3),x, algorithm="fricas")

[Out]

1/4*sqrt(3)*log((2*x^2 + sqrt(3)*(2*x + 3) + 6*x + 6)/(2*x^2 + 6*x + 3)) + 1/4*log(2*x^2 + 6*x + 3)

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Sympy [A]  time = 0.09926, size = 46, normalized size = 0.94 \begin{align*} \left (\frac{1}{4} - \frac{\sqrt{3}}{4}\right ) \log{\left (x - \frac{\sqrt{3}}{2} + \frac{3}{2} \right )} + \left (\frac{1}{4} + \frac{\sqrt{3}}{4}\right ) \log{\left (x + \frac{\sqrt{3}}{2} + \frac{3}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2*x**2+6*x+3),x)

[Out]

(1/4 - sqrt(3)/4)*log(x - sqrt(3)/2 + 3/2) + (1/4 + sqrt(3)/4)*log(x + sqrt(3)/2 + 3/2)

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Giac [A]  time = 1.05922, size = 62, normalized size = 1.27 \begin{align*} -\frac{1}{4} \, \sqrt{3} \log \left (\frac{{\left | 4 \, x - 2 \, \sqrt{3} + 6 \right |}}{{\left | 4 \, x + 2 \, \sqrt{3} + 6 \right |}}\right ) + \frac{1}{4} \, \log \left ({\left | 2 \, x^{2} + 6 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2*x^2+6*x+3),x, algorithm="giac")

[Out]

-1/4*sqrt(3)*log(abs(4*x - 2*sqrt(3) + 6)/abs(4*x + 2*sqrt(3) + 6)) + 1/4*log(abs(2*x^2 + 6*x + 3))